Engineering topics
Publications and source records attributed to Johnson, I. L., Jr..
A process for adjusting perturbation magnitude for accurate difference-quotient estimation of derivatives is described. The process is intended to be carried out sequentially, alternating with iterations of a parameter-optimization algorithm. A more complex and computationally-expensive scheme for occasional auxiliary use is also described.
Gradient projection is a powerful algorithm for minimization of a function subject to constraints. Constraint nonlinearities hamper projection computations. The constraints must then be restored before another projection cycle. The restoration steps taken in the process of following nonlinear constraint surfaces can be used as a guide to the construction of a curve which more nearly follows the constraints than does the straight line in the projected gradient direction. This scheme, termed 'curvilinear projection', was explored in earlier research. The study presently reported carries out some computational experiments using a related version of the technique. Some other details of projection computations which turn out to be practically important are taken up: rules for updating the variable metric in projection when early termination of the one-dimensional search on constraint violation occurs, and active-constraint logic for screening inequalities that makes use of the Kuhn-Tucker necessary conditions.
An optimal solution of the ascent trajectory of the Space Shuttle for the first orbital flight test is presented; the optimization is a minimum propellant, four-control problem in yaw angle, roll angle, pitch angle and vacuum thrust of each Space Shuttle main engine. Piecewise linear segments with juncture points treated as parameters are employed to model the controls. Equations of motion for a three-dimensional flight with pitch plane moment balance about an oblate are integrated numerically with a fourth-order Runge-Kutta method; two- and one-dimensional cubic spline function curve fits of aerodynamic coefficients are used during the first and second stages, respectively. The constraint minimization problem is solved with the Davidon-Fletcher-Powell function method.
The Fletcher-Powell version of the Davidon variable metric unconstrained minimization technique is described. Equations that have been used successfully with the Davidon-Fletcher-Powell penalty function technique for solving constrained minimization problems and the advantages and disadvantages of using them are discussed. The experience gained in the behavior of the method while iterating is also related.
A variable-metric algorithm is described that uses both linear and quadratic penalty terms for handling nonlinear constraints. Quadratic penalty coefficients are adjusted in a process which maintains a positive-definite matrix of second partial derivatives of the function without generating the large positive eigenvalues which cause zigzagging and slow convergence. The schemes suggested use inferred second-order properties not only in terms of the variable metric of the Davidson-Fletcher-Powell algorithm (or its relatives) but by estimating of second directional derivatives by fitting cubics to various functions along search directions.
In the search technique presented, the number of function evaluations is reduced by making use of the derivative of the function. The search method is combined with the variable metric algorithm reported by Davidon (1959) to solve a 17-parameter optimal atmospheric flight of a space shuttle vehicle. In the case of the problem, the proposed method requires one or two function evaluations per iteration less than the standard cubic fit-golden section method.
Conjugate direction minimization procedure based on sequences and inferred second partial derivative matrix inverse
Impulsive orbit transfer optimization using accelerated gradient program based on Newtonian algorithm for digital computer method
Impulsive orbit transfer optimization using accelerated gradient method